Aiden takes out a 30 year loan for $24,000 to be repaid with payments at the end of each year consisting of interest on the loan and a sinking fund deposit. Interest is charged at a 16% annual rate. The sinking fund’s annual rate is 11%. However, beginning in the 13th year, the annual effective interest rate on the sinking fund drops to 8%. As a result, the payments are increased by X.

Calculate X.

• $228.01
• $348.60
• $447.12
• $273.41
• $337.67

Hardiek, Aaron (June 2010). "Study Questions for Actuarial Exam 2/FM". digitalcommons.calpoly.edu. Retrieved November 20, 2023.

Solution: A

[[math]] \begin{aligned} & \mathrm{OB}_0=24,000 \\ & \mathrm{i}=.16 \end{aligned} [[/math]]

j on sinking fund: .11 first 12 years then .08 for last 18 years

Original payments would be:

[[math]] \begin{aligned} & \mathrm{Ks}_{\overline{30}|0.11}=24,000 \\ & \mathrm{~K}=120.59 \end{aligned} [[/math]]

At the end of 12 years:

[[math]] 120.50 \mathrm{~s}_{\overline{12}|0.11}=2,738.98 [[/math]]

With the new rate of interest, payment increases to: [math]120.59+\mathrm{x}[/math]

The accumulated value is;

[[math]] \begin{aligned} & 2,738.98(1.08)^{18}+(120.59+\mathrm{x}) \mathrm{s}_{\overline{18}|0.08}=24,000 \\ & (120.59+\mathrm{x}) \mathrm{s}_{\overline{18} | 0.08}=13,054.98 \\ & \mathrm{x}=228.01 \end{aligned} [[/math]]

Hardiek, Aaron (June 2010). "Study Questions for Actuarial Exam 2/FM". digitalcommons.calpoly.edu. Retrieved November 20, 2023.